The author studies elliptic equations of the type \(\text{div}(a(x,u,Du))=\text{div}(g)\), \(u\in W^{1,(q_i)}_0(\Omega)\), and integral functionals of the calculus of variations of the form \(F(u)=\int_\Omega f(x,Du)dx\) when \(a\) or \(f\) satisfy the anisotropic coerciveness condition \(f(x,\xi)\geq m\sum^N_{i=1}|\xi_i|^{q_i}\). Various regularity results are obtained in Lorentz spaces, Orlicz spaces and in the Lebesgue space \(L^\infty(\Omega)\). The estimates are obtained by using test functions defined thanks to the nonincreasing rearrangement of \(u\). These results extend previous results obtained in this direction [cf., e.g., N. Fusco and C. Sbordone, Commun. Partial Differ. Equations 18, No. 1-2, 153-167 (1993; Zbl 0795.49025); B. Stroffolini, Boll. Unione Mat. Ital., VII. Ser. A5, No. 3, 345-352 (1991; Zbl 0754.46026)].